Many-body effect determines the selectivity for Ca2+

and Mg2+ in proteinsZhifeng Jinga, Chengwen Liua, Rui Qia, and Pengyu Rena,1

aDepartment of Biomedical Engineering, The University of Texas at Austin, Austin, TX 78712

Edited by Pavel Jungwirth, Academy of Sciences of the Czech Republic, Prague, Czech Republic, and accepted by Editorial Board Member Peter J. Rossky July 3,2018 (received for review March 24, 2018)

Calcium ion is a versatile messenger in many cell-signaling pro-cesses. To achieve their functions, calcium-binding proteins selec-tively bind Ca2+ against a background of competing ions such asMg2+. The high specificity of calcium-binding proteins has been in-triguing since Mg2+ has a higher charge density than Ca2+ and isexpected to bind more tightly to the carboxylate groups in calcium-binding pockets. Here, we showed that the specificity for Ca2+ isdictated by the many-body polarization effect, which is an energeticcost arising from the dense packing of multiple residues around themetal ion. Since polarization has stronger distance dependence com-pared with permanent electrostatics, the cost associated with thesmaller Mg2+ is much higher than that with Ca2+ and outweighs theelectrostatic attraction favorable forMg2+. With the AMOEBA (atomicmultipole optimized energetics for biomolecular simulation) polar-izable force field, our simulations captured the relative bindingfree energy between Ca2+ andMg2+ for proteins with various typesof binding pockets and explained the nonmonotonic size depen-dence of the binding free energy in EF-hand proteins. Without elec-tronic polarization, the smaller ions are always favored over largerions and the relative binding free energy is roughly proportional tothe net charge of the pocket. The many-body effect depends onboth the number and the arrangement of charged residues. Fine-tuning of the ion selectivity could be achieved by combining themany-body effect and geometric constraint.

metalloproteins | EF hands | polarizable force fields | free-energycalculation

Metal ions are essential for a variety of biological functionssuch as homeostasis, muscle contraction, and enzyme ca-

talysis (1–3). For example, Ca2+ acts as a second messenger thatcontrols many cellular processes by inducing conformationalchanges of the receptor proteins. Ca2+ signaling is usedthroughout the life cycle of an organism, including proliferation,metabolism, and cell death (4, 5). Under certain conditions, el-evated Ca2+ levels are cytotoxic. Thus, Ca2+ concentration mustbe strictly maintained within spatial and temporal boundaries (4,5). Mg2+ is crucial in energy-requiring metabolic reactions sinceATP must be bound to Mg2+ to be biologically active (2, 6).Mg2+ also forms a component of RNA and DNA tertiary struc-tures, and is necessary for the proper structure and activity ofRNA and DNA polymerases (2).The ability of proteins to select a specific metal ion over

similar ones is critical to these cellular processes. Many effortshave been made to understand the selectivity of proteins formetal ions (7–12). One well-studied case over the past decades ispotassium channel. The selectivity of potassium channels for K+

over Na+ has been attributed to the intrinsic electrostatic prop-erties of the pores, geometric flexibility in the structural aspects,and kinetic effects (7, 11–13).Another puzzling pair of metal ions is Ca2+ and Mg2+. Mg2+ is

a better charge acceptor due to its smaller size and higher chargedensity, so it would be expected to bind more tightly to carbox-ylate groups commonly found in calcium-binding motifs (7, 14),if the energy cost of ion dehydration is omitted. Counterintui-tively, EF hands (named after the E and F helices of parvalbumin)

typically have 102- to 104-fold higher affinity for Ca2+ (15). Twopopular explanations for the EF-hand selectivity are (i) the pen-tagonal bipyramidal geometry and relatively large size preferablefor Ca2+ and (ii) the larger solvation penalty of Mg2+ (7, 16). Thegeometric explanation is closely related to the snug-fit mechanismfor ion channels and classical concepts in host-guest chemistry (13,17). However, mutants of EF-hand protein with octagonal geom-etry also favor Ca2+ (albeit with weaker selectivity) (18), whichindicates that the bipyramidal geometry is not essential. On theother hand, the solvation penalty argument is more of a phenom-enological explanation; the question remains as to why the stronginteraction of Mg2+ with charged carboxylate groups in the bindingpocket cannot overcome the relatively weak interaction with water.In addition, EF-hand proteins have high specificity for Ca2+ overMg2+, Sr2+, and Ba2+ (19), while the CheY protein, also withmultiple carboxylate groups in the binding pocket, has very weaksize selectivity (20). These different patterns of size selectivitycannot be attributed to the solvation penalty. In other words, fine-tuning of selectivity of Ca2+ vs. Mg2+ has to come from the inter-action with the binding pockets rather than solvation.There have been insightful studies on the determinants of

calcium selectivity. The effect of the number of negativelycharged residues has been studied both experimentally andcomputationally, and mechanisms based on the electrostatic re-pulsion have been proposed (21–23). The geometric constraintor deformation (24–26) is a frequently discussed factor for sizeselectivity. Lim and coworkers (7, 9, 27, 28) showed that thesecond shell and the protein environment either stabilize orenhance the properties of the inner shell. The roles of polari-zation (29–34) and quantum effects (26, 35, 36) have also been

Significance

Metal ions have important biological functions and are associ-ated with diseases including cancer and neurodegenerative dis-orders. The fundamental question of metal ion selectivity inproteins has received continued interest over the past decades.Compared with Na+/K+, the selectivity for Mg2+/Ca2+ is less wellunderstood. Although Mg2+ is a better charge acceptor, calcium-binding proteins with highly charged binding pockets can se-lectively bind Ca2+ against a much higher concentration of Mg2+.Here we show that this selectivity is dictated by the many-bodypolarization effect, which is a cost arising from the dense pack-ing of multiple residues around the metal ion. By combininggeometric constraint and the many-body effect, it is possible tofine-tune the selectivity for metal ions of different sizes.

Author contributions: Z.J. and P.R. designed research; Z.J. performed research; Z.J. ana-lyzed data; and Z.J., C.L., R.Q., and P.R. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. P.J. is a guest editor invited by the Editorial Board.

Published under the PNAS license.1To whom correspondence should be addressed. Email: pren@mail.utexas.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1805049115/-/DCSupplemental.

Published online July 23, 2018.

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pointed out. Nevertheless, the preference for Ca2+ can be foundin many families of proteins (7, 19, 37–41), including an Mg-dependent enzyme ribonuclease H1 (42), while Mg-specificproteins are less common. A general understanding of thehigher affinity for Ca2+ in different proteins is still lacking.In this work, we first used ab initio calculations on model

compounds of the binding pockets to illustrate the importance ofthe many-body effect on Ca2+ selectivity. Then, through freeenergy calculations with a polarizable force field for six calcium-and magnesium-binding proteins, we demonstrated that themany-body effect depends on both the composition and the ge-ometry of the binding pocket. Further, we showed how proteinscan precisely control the selectivity by utilizing many-body effectand geometric constraint, leading to nonmonotonic size de-pendence of ion selectivity.

Results and DiscussionDue to its smaller size, Mg2+ interacts much more strongly withacetate in gas phase than Ca2+ does compared with the corre-sponding ion–water interactions (Fig. 1). At typical separationdistances in crystal structures, the difference between the Mg2+-acetate and Ca2+-acetate dimer interaction energies is ∼40 kcal/mol,significantly larger than the difference of ∼25 kcal/mol for theion–water dimer. EF-hand binding pockets have three to four

carboxylate groups (Asp/Glu) (16) together with carbonyl andhydroxyl groups, and the coordination numbers for Mg2+ and Ca2+

are generally the same as those in water (6 for Mg2+ and 7 for Ca2+).If all these interactions between metal ions and individual func-tional groups are additive, EF-hand binding pockets would fa-vorably bind Mg2+ by a large margin.In solution, Mg2+ and Ca2+ have comparable binding free

energies with acetate (−1.7 vs. −1.6 kcal/mol) (43). In fact,chemical concepts from solution-phase experiments indicate thatMg2+ and Ca2+ are similar in binding abilities. The hard-ligandscale of Mg2+ is barely larger than that of Ca2+ (i.e., the twometal ions have similar binding affinities for hard ligands) (44).In Hofmeister series, Mg2+ and Ca2+ are close to each otherwhile their order varies depending on the solute (41, 45–47). Thesimilarity between Mg2+ and Ca2+ binding affinities may beexplained by the dielectric screening of the solvent (48). How-ever, multivalent ligands such as EDTA (49) and many proteins(7, 19, 37–42) tend to have higher affinities for Ca2+. Thesestrong preferences for Ca2+ must be due to the cooperative ormany-body effect of the ligands around the ions (50).To illustrate the many-body effect in ion binding, we calculated

the quantum-mechanical (QM) interaction energies for modelcompounds of protein–ion complexes and conducted many-bodyenergy decomposition analysis (Table 1). Three EF-hand proteinswith both Ca2+- and Mg2+-bound structures available in the Pro-tein Data Bank (PDB) were chosen. The model systems wereconstructed by taking the first shell chemical groups (includingwater) around the ion out of the PDB structures. By definition, thetotal interaction energy is the sum of two-body and many-bodyenergies, where the two-body energy is the sum of interactionenergies of all pairs of molecular fragments in the system. Similarto the dimer interaction energies in Fig. 1, the two-body energiesfor Ca2+ are much weaker than those for Mg2+, as evidenced bythe large positive numbers of ΔΔE2B(Mg2+ → Ca2+). However,the large difference in the repulsive many-body interactions favorsCa2+, which compensates for its unfavorable two-body interac-tions. After incorporating the contribution of many-body interac-tions, the gaps between Mg2+ and Ca2+ interaction energies,ΔΔEint(Mg2+ → Ca2+), are brought down to ∼70 kcal/mol, whichapproaches the difference in experimental hydration free energy(∼78 kcal/mol). It is noteworthy that the two-body interactionshere include both ion–ligand attraction and ligand–ligand re-pulsion, so the two-body repulsive interaction between ligandsis not a decisive factor for the observed selectivity. We estimatedthe relative free energies using the total interaction energies ofthe model systems and the experimental hydration free energies(Table 1). The calculated ΔΔGbind(Mg2+ → Ca2+) values of all ofthe three EF-hand systems are lower than or close to 0, meaningthat the binding pockets would prefer Ca2+. Without the many-body interactions, the predicted free energies would strongly favorMg2+, which conflicts with experiments. Alternative analyses usingthe same binding pocket geometry for Mg2+ and Ca2+ or usingdifferent reference hydration free energies also show that many-body interactions strongly favor Ca2+. (See SI Appendix for de-tails.) The calculations with the simple model systems suggest that

Fig. 1. Comparison of ion–acetate and ion–water dimer interaction ener-gies. (A) Mg2+/Ca2+ and acetate; (B) Mg2+/Ca2+ and water; (C) differencebetween the interaction energies of Mg2+ and Ca2+. The ions were placed onthe bisector of the O-C-O or H-O-H angle. The interaction energies werecalculated by RI-MP2/def2-QZVPPD. The shaded areas in A and B indicatetypical distances in PDB or aqueous solution. ΔEint, interaction energy; Ac,acetate; Ang., angstrom.

Table 1. Comparison of relative binding free energies from QM calculation and experiment

PDB Model compounds ΔΔE2B,M-L ΔΔE2B,L-L ΔΔE2B ΔΔEMB ΔΔEint Calc. ΔΔGbind Expt. ΔΔGbind

1IG5/4ICB Ac3B2W2 251.9 −19.3 232.6 −165.6 67.0 −10.8 −6.22LVJ/K Ac4B1W1 160.9 −16.6 144.3 −73.1 71.1 −6.7 −2.41B8L/C Ac4B1W1 217.8 −37.5 180.3 −103.4 77.0 −0.8 −1.6

Reported values are the differences (in kcal/mol) between Mg complexes and Ca complexes, ΔΔE = ΔE (Ca2+) − ΔE (Mg2+). ΔΔGbind is approximatelycalculated by ΔΔGbind = ΔΔEint − ΔΔGsolv

Expt = ΔΔEint − 77.8 kcal/mol. The total interaction energy is calculated by ΔEint = Etotal − Σi Ei, where Ei is the energy ofisolated monomer i. The two-body interaction energy is calculated by ΔE2B = Σi, j (Eij − Ei − Ej), where Eij is the energy of isolated dimer. ΔΔE2B,M-L and ΔΔE2B,L-Lare the two-body interaction energies between the metal ion and ligands and between ligands and ligands, respectively. The many-body interaction energy isΔEMB = ΔEint − ΔE2B. Ac, B, and W represent acetate, acetamide, and water, respectively. Calc., calculated; Expt., experiment; int, interaction; solv, solvation.

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the many-body interactions in the first shell are crucial for explainingCa2+ selectivity.It should be emphasized that because of the approximations

made in the above calculations (e.g., simplified model compounds,use of static structures, lack of entropy contribution, the choices ofsolvation free energies, and errors of QM methods; see SI Ap-pendix for detailed discussion), the apparently good agreementbetween calculated and experimental free energies seems fortu-itous. Therefore, the results can perhaps only be used to dem-onstrate the qualitative effect of the many-body interactions. Tocalculate the free energies rigorously, the effect of protein envi-ronment and conformational sampling at finite temperatureshould be considered [e.g., through molecular dynamics (MD)simulations]. Due to their computation cost, MD simulations areusually performed with empirical force fields. The large, repulsivemany-body energy in Table 1 is not explicitly modeled in additiveforce fields, while it is included in polarizable force fields since alarge portion of the many-body energy comes from polarization(51, 52). A comparison between many-body energies fromMP2 and the AMOEBA (atomic multipole optimized energeticsfor biomolecular simulation) force field for several model com-pounds of protein–metal ion complexes (see SI Appendix, Fig. S2for structures), including both EF hand and non-EF hand, isshown in Fig. 2 and SI Appendix, Table S3. The many-body energyincreases with the number of carboxylate groups, and Mg2+

complexes have larger many-body energies than Ca2+ systems.Although, compared with MP2, AMOEBA systematically under-estimates the many-body energies by ∼30 kcal/mol for both Mg2+

and Ca2+, a strong correlation is observed. The systematic error ofAMOEBA should not have a significant impact on the calculationof relative affinity. The AMOEBA total interaction energies forMg2+ complexes are noticeably weaker than MP2 energies.The importance of the many-body interactions motivated us to

investigate the binding between different proteins and Mg2+/Ca2+

ions through MD simulations. Fig. 3A shows the calculatedand experimental relative binding free energy ΔΔGbind(Mg2+ →Ca2+). Since the many-body effect is modeled by polarization inthe force field, we also performed nonpolarizable simulations forcomparison (Fig. 3A). The nonpolarizable simulations used ei-ther AMOEBA with polarization turned off or the fixed-chargeAMBER (assisted model building with energy refinement) forcefield. Six proteins were studied: namely, EF-hand proteins par-valbumin (5CPV and 1B8L) (18) and calbindin D9k (4ICB) (39),a carboxylate cluster Mg-binding protein CheY (20), theC2 domain of dysferlin (4IHB) (38), and an inserted (I) domainof integrin (1ZOO) (53). A description of these proteins and thestructures of the binding pockets are provided in Fig. 4 and Table2. While ΔΔGbind(Mg2+ → Ca2+) values are systematicallyunderestimated by ∼2 kcal/mol by AMOEBA (Fig. 3A), which is

likely because the Mg2+ interaction is not strong enough (Fig. 2),the trend across all six proteins is reproduced. The bindingpockets with more Asp/Glu tend to prefer Ca2+, while the se-lectivity is not solely determined by the net charge of the pocket.For example, 1B8L and 2CHE are highly charged but have rel-atively weak selectivity. Both trends can be explained by themany-body effect. When there are more charged residues, therepulsive many-body effect is more significant and outweighsthe favorable pairwise interactions with individual side chains.1B8L and 2CHE have a relatively weaker many-body effect dueto their pocket geometries: 1B8L is a triple mutant of 5CPV inwhich the last residue Glu101 of the EF-hand motif is replacedby Asp, which creates a larger binding pocket and reduces themany-body effect. In 2CHE, the charged residues are clusteredin one hemisphere so that they cannot all tightly bind to themetal ion (Fig. 4), which also reduces the many-body response ofthe binding pocket. This explanation is confirmed by the relativebinding enthalpies in Fig. 3B. Among the four proteins withthree or more charged residues, 5CPV and 4ICB have favorablepolarization enthalpies for Ca2+ binding, while 1B8L and 2CHEhave relatively unfavorable polarization enthalpies. In theAMOEBA nonpolarizable simulations, all ΔΔGbind(Mg2+ →Ca2+) values become positive and roughly proportional to thenumber of Asp/Glu. This coincides with the naive intuition thatproteins with more negative charges should bind more strongly toMg2+ but is contrary to the results of polarizable MD simulationsand experiments. For pockets with the same number of Asp/Glu, therelative binding free energies are similar in AMOEBA nonpolariz-able simulations but variable in polarizable simulations. Clearly,polarization is more sensitive to the pocket geometry than perma-nent electrostatic interaction. In AMBER simulations, the correla-tion between ΔΔGbind(Mg2+ → Ca2+) and the number of Asp/Glu is weaker, possibly due to the partial incorporation ofpolarization in atomic charge parameters (54, 55).

Fig. 2. Comparison between MP2 and AMOEBA many-body energies andtotal interaction energies of the model compounds. Data are tabulated in SIAppendix, Table S3.

Fig. 3. Comparison between experimental and calculated relative bindingfree energies. (A) Experimental and calculated free energies as a function ofthe number of carboxylate groups. The SDs of all calculated free energies are0.2 kcal/mol. The color codes represent different proteins and are the sameas used in B. (B) Polarization contribution to relative binding enthalpy inAMOEBA results. For most of the binding pockets, the relative binding en-thalpy and the polarization contribution are negative. (C) Correlation betweenexperimental binding free energies and calculated values using AMOEBA orAMBER/MDEC. The Pearson correlation coefficients are 0.97 and 0.63 forAMOEBA and AMBER/MDEC, respectively. Data are tabulated in SI Appendix,Tables S4 and S5. Calc., calculated; Expt., experiment; POL, polarization.

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We also explored whether the polarization effect can bemodeled by a dielectric continuum, as proposed by Leontyev andStuchebrukhov (56). In their MD in electric continuum (MDEC)method, the atomic charges of ionized groups are scaled by 0.7,corresponding to a uniform effective dielectric constant of ∼2.The calculated binding free energies (Fig. 3C) indeed show asignificant improvement over the results of unscaled AMBERforce field (Fig. 3A). However, MDEC gives a weaker correla-tion with experimental relative binding free energy (R2 = 0.63)than the fully polarizable AMOEBA force field does (R2 = 0.97)across the six proteins. MDEC also predicts the inconsistentranking of selectivity for Mg2+-binding proteins 1ZOO and2CHE and the EF-hand mutant 1B8L. These results are notsurprising as the local polarization depends on the detailedchemical composition and geometry while MDEC assumes auniversal dielectric constant.The striking difference between polarizable and nonpolariz-

able simulations reveals the effect of polarization on Ca2+ se-lectivity. Polarization energy can also be decomposed into two-body and many-body contributions. In the classical AMOEBAmodel, the many-body effect arises solely from polarization, so

the many-body energy is the same as many-body polarizationenergy. Since Mg2+ has higher charge density and shorter co-ordination distances, the two-body polarization strongly favorsMg2+ binding over Ca2+ binding (see SI Appendix, Table S2 forenergy decomposition analysis of model compounds). Therefore,we may also argue that it is the many-body (polarization) effect,rather than two-body polarization, that dictates the Ca2+

selectivity.To understand the polarization effect, we can write the po-

larization energy as the sum of dielectric screening energy andself-energy or Born energy, EPOL = Escreen + EBorn (56). TheBorn energy reflects the relaxation of the electronic degrees offreedom in the environment, which is always negative. Thescreening energy reduces the magnitude of electrostatic in-teraction, and is thus positive in most cases. The total polariza-tion energy is always negative. If only two-body interaction isconsidered (i.e., the interaction between a pair of molecularfragments is independent of the polarization of others), therewill be no screening of electrostatic interaction and the Bornenergy will dominate. Consequently, the two-body polarizationin the highly charged binding pocket is stronger than in ion hy-dration (Fig. 5A). After including the many-body polarization,there will be a large screening energy in the binding pocket be-cause of the strong electrostatic interactions. Therefore, the totalpolarization energy in the binding pocket becomes less favorablethan that in ion hydration, which can be seen in the results ofboth gas-phase cluster calculation and condensed-phase MDsimulations (Fig. 5 A and B). In general, the binding pockets withmore carboxylate groups (e.g., 5CPV) have more unfavorablepolarization enthalpy than pockets with fewer carboxylate groups(e.g., 2CHE and 1ZOO). The ion selectivity for Ca2+ vs. Mg2+ ismainly caused by the different distance dependence of electro-static and polarization energies. As the ion size increases andion-residue distances increase accordingly, the magnitudes of thefavorable permanent electrostatic interaction and the many-bodypolarization penalty both decrease, while the latter decreasesmuch faster (Fig. 5C and SI Appendix, Fig. S3). Thus, the overallbinding energy becomes more favorable with increasing ion size.For large ions, the short-ranged polarization contribution grad-ually disappears and the change in total binding energy wouldslow down or even reverse direction.To examine whether the simple rule of ion selectivity stated

above applies to divalent ions beyond Mg2+ and Ca2+, we con-ducted an alchemical experiment in which the size of the metalion was varied from that of Mg2+ to roughly that of Ba2+. It isknown that Mg2+ has stronger polarizing ability than Ca2+ at thesame separation distance, while polarizabilities of the metal ionsare negligible (57). To focus on the effect of ion size, the po-larization parameter of Mg2+ in our model was used for all ionsin this series (see SI Appendix, Table S7 for force-field param-eters). Relative binding free energies for these ions are plotted inFig. 6. The six proteins show different patterns of dependence on

Fig. 4. Structures of the binding pockets from MD simulations. (A–F) 5CPV,1B8L, 4ICB, 2CHE, 4IHB, and 1ZOO bound with Mg2+, respectively. A–C areEF-hand proteins.

Table 2. Description of proteins studied in this work

PDB Protein Function Ion

Expt. ΔGbind

(kcal/mol)

No. of COO− in first/second shellMg2+ Ca2+

5CPV Carp parvalbumin Muscle contraction Ca2+ −5.4 −11.0 4/01B8L D51A/E101D/F102W mutant of 5CPV Ca2+ −6.7 −8.3 4/04ICB Bovine calbindin D9k Calcium transport Ca2+ −3.0 −9.2 3/02CHE Response regulator protein CheY Bacterial chemotaxis Mg2+ −4.0 −4.5 2/14IHB C2A domain from human dysferlin Membrane repair Ca2+ −5.7 −7.4 2/01ZOO I domain from the CD11a integrin Leukocyte adhesion Mg2+ −6.4 −4.7 1/1

Expt., experiment.

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the ion size. Among the three EF-hand proteins, 4ICB favors thelargest ion, while 5CPV and 1B8L have an optimal ion sizearound vdW (van der Waals) diameter σ = 3.6 Å. These trendsare related to competition between electrostatic and polarizationdiscussed above as well as the flexibility of the binding pockets.4ICB has a more flexible binding pocket: the carbonyl groupfrom Gln22 is available for coordinating large ions, allowing thecoordination number to increase to 8 for the largest ion (Fig.6D), close to the coordination number of 9 in water. Similarly,the binding free energy of the C2 domain 4IHB decreasesmonotonically with ion size, since the binding pocket is formedby flexible loops and there is no constraint on the pocket ge-ometry. The other two EF hands, 5CPV and 1B8L, have exten-sive hydrogen-bond networks and rigid binding pockets (Fig. 4),which restrict the coordination number to ∼7 (SI Appendix, Fig.S4). This causes both strain energy and a loss in electrostaticinteraction for large ions compared with the binding with flexiblepockets, and consequently an upward trend of the binding freeenergy. As discussed earlier, the relatively small many-body effectin 2CHE binding pocket leads to a weaker selectivity for largeions. 1ZOO is the only protein with positive relative binding freeenergy for the largest ion, which results from a combination of itssmall many-body effect and rigid binding pocket.The presence of an optimal ion size for the EF-hand proteins

5CPV and 1B8L and the weak selectivity of 2CHE for large-sized

ions agree qualitatively with experimental data for an EF-handlike protein, D-galactose and D-glucose receptor (GGR),and the CheY protein (Fig. 6C) (19, 20). It should be noted thatsince the simulated ions have stronger polarization comparedwith their real-world counterparts, the binding free energiesrelative to that of Mg2+ would be more negative than those ofreal ions. Although there is no direct experimental data tocompare with the simulation results of the bovine calbindin D9k(4ICB), it has been shown that neutral substitution at the gate-way position of an EF hand retains the selectivity for Ca2+ overMg2+ but leads to a partial loss of selectivity over larger ions(22). This is consistent with our results—that is, when the ge-ometry of the binding pockets is less well constrained, thebinding affinities for larger ions will be enhanced.The contribution of polarization can be clearly seen by

comparing the results of polarizable and nonpolarizable simu-lations (Fig. 6B). In nonpolarizable simulations, the bindingfree energies of all proteins increase with the ion size. 5CPVand 1B8L have the steepest slopes due to their strong geometricrestraint and high number of negative charges. Without polar-ization, the diversity of ion selectivity in proteins would begreatly reduced.

Fig. 5. Role of polarization in ion selectivity. (A) Energy changes betweenMg2+(H2O)6 cluster and Mg2+(Ac−)4 cluster. POL and NP are the polarizationand nonpolarization contributions. NP is defined as the sum of all energyterms in AMOEBA except polarization, including intramolecular bondedenergy, van der Waals, and permanent electrostatics energies. In ionic in-teractions, the NP term is dominated by electrostatics. The Mg2+(H2O)6cluster has more favorable two-body polarization energy but less favorabletotal polarization energy. (B) Polarization and nonpolarization enthalpychanges ΔH for Mg2+

–protein binding. The same as in A, Mg–proteinbinding has favorable ΔH compared with hydration but less favorable po-larization enthalpy. (C) Illustration of energy changes due to polarizationand permanent electrostatics (ELST) for transferring aqueous metal ion toprotein. Protein–ion binding is driven by electrostatics and must overcomethe polarization penalty. As the ion size increases, both permanent elec-trostatic interaction and polarization penalty become weaker, while thepolarization penalty decreases much faster. As a result, larger ions will befavored. Ac, acetate.

Fig. 6. Relative binding free energy as a function of vdW size. (A) Calcu-lated relative binding free energies using AMOEBA; (B) calculated relativebinding free energies without polarization; (C) experimental binding freeenergies for the D-galactose and D-glucose receptor (GGR) and the CheYprotein 2CHE (19, 20); (D) ion coordination number in AMOEBA simulations;(E) structures of 4ICB bound to Mg2+ (Left) and the largest hypothetical ion(Right). 5CPV, 1B8L, and 4ICB in A and GGR in C have EF-hand motifs. Theions in the simulations all have +2 charges and the same polarization pa-rameters as Mg2+. All of the calculated binding free energies are relative tothat of Mg2+, which has a vdW size of 2.90 Å. σ = 3.59, 4.03, and 4.45 Åroughly correspond to Ca2+, Sr2+, and Ba2+, respectively. The SDs of all cal-culated results are ∼0.2 kcal/mol. Ang., angstrom; Coord. No., coordinationnumber; Expt, experiment.

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ConclusionWe have shown that the selectivity of proteins for Ca2+ overMg2+ originates from the energetic cost due to many-bodyinteractions, which can be captured with a classical polarizationmodel based on self-consistent atomic dipole induction. Thebinding of the larger ion Ca2+ is associated with lower cost thanthe smaller Mg2+, and thus has higher binding affinities. Oursimulations with a polarizable force field could reproduce therelative binding energies between Mg2+ and Ca2+ across varioustypes of binding pockets. Without polarization, Mg2+ will befavored by all highly charged binding pockets. The many-bodypolarization depends on the number of charged residues sur-rounding the ion and is sensitive to their spatial arrangement,which can be used to fine-tune the selectivity. We also demon-strated that the nonmonotonic ion size dependence of bindingfree energies for different proteins arises from the competitionsamong permanent electrostatic, many-body polarization andgeometric restraints. In particular, the precise control of ion sizeselectivity in EF hands is achieved by combining geometricconstraint and the many-body effect.As general principles, Mg-binding proteins should have more

polar and fewer charged groups (to reduce the polarizability andelectrostatic interaction) or suboptimal geometries that weakenthe electrostatic interaction. Ca-binding proteins should havemultiple charged residues and rigid geometry; highly chargedand flexible binding pockets would favor large divalent ions.

Materials and MethodsModel compounds for QM calculations were extracted from three proteinswith both Mg-bound and Ca-bound structures in PDB (i.e., 1IG5/4ICB, 2LVJ/2LVK, and 1B8L/C). The first-shell carboxylate, carbonyl, and hydroxyl groupswere represented by acetate, acetamide, and ethanol, respectively. The firstshell is defined as functional groups that directly coordinate the metal ion,and the second shell interacts with the first shell but not in contact with themetal ion. Second-shell Asp or water in 2LVJ, 1IG5, and 4ICB was also includedto ensure the same number of molecules in Mg2+ and Ca2+ complexes. Thepositions of all of the nonhydrogen atoms were fixed at their respective PDBcoordinates, while the hydrogen positions were optimized at the B3LYP/6–31G(d) level of theory. The many-body energy decomposition for these

model compounds was calculated at the RI-MP2/def2-QZVPPD level. Con-sidering the large size of the basis set, no correction for basis-set superpo-sition error was applied. The Gaussian 09 (58) and the Psi4 package (59) wereused for the geometry optimization and energy calculations, respectively.

The AMOEBA free energy simulations were performed using the TINKER-OpenMM package (60). The latest AMOEBA parameter was used (51, 61, 62),which includes modifications detailed in SI Appendix. The parameter file inTINKER format is also included in Dataset S1. The absolute binding freeenergies of acetate computed by AMOEBA were −2.7 ± 0.2 and −2.5 ±0.2 kcal/mol for Mg2+ and Ca2+, respectively; for comparison, the corre-sponding experimental values are −1.7 and −1.6 kcal/mol (43). For the arti-ficial ions, the vdW parameters were obtained by linear interpolation usingMg2+ and Ca2+ parameters and experimental ionic radii (63). The Bennetacceptance ratio method was employed to analyze the free energy differ-ences. The system set-up was as follows. The hydrogen of each protein wasassigned at pH 7 by using PDB2PQR 2.0.0. Then, the protein was solvated in aperiodic cubic box with the edge length ≥30 Å larger than the longest di-mension of the protein. Sodium and chloride ions were added to neutralizethe charge and to give a salt concentration of 150 mM. The systems wereminimized, equilibrated at 100 K for 100 ps with protein position fixed, andgradually heated from 100 to 300 K in 1.6 ns. Then, the box volumes weredetermined by a 1-ns NPT (isothermal–isobaric ensemble) simulation at 300K and 1 bar. In free energy perturbation, Mg2+ was gradually changed toCa2+ in five steps, with all force-field parameters linearly interpolated. A 6-nsNVT (canonical ensemble) simulation at 300 K for each alchemical state wascarried out with a 3-fs integration time step and coordinates were savedevery 3 ps for analysis. Hydrogen-mass repartitioning was applied to allowthe large time step. Additional simulation parameters, including the vdWcutoff, the polarization PME parameters, and thermostat, are same as in ourprevious work (60) (Dataset S2). The AMBER and AMBER/MDEC simulationswere performed using the PMEMD CUDA program in Amber 16 package (64,65), with the ff14SB protein force field (66) and the HFE set of ion parametersfrom Li et al. (67). All of the bonds involving hydrogen were constrained by theSHAKE algorithm. The integration time step was 2 fs and temperature wascontrolled by the Langevin thermostat. For each alchemical state, a 10-ns NVTtrajectory was collected with an interval of 5 ps. The system setup and analysiswere identical to those of the AMOEBA simulations.

ACKNOWLEDGMENTS. We thank Mr. Piao Ma for critical reading of themanuscript. We are grateful for support by the NIH (Grants R01GM106137and R01GM114237), the Robert A. Welch Foundation (Grant F-1691), andCancer Prevention Research Institute of Texas (Grant RP160657).

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